Base change/Transformation matrix/2x2-matrices/1/Exercise

Let ${}V$ be the vector space of the ${}2\times 2$-matrices with the standard basis

${}{\mathfrak {u}}={\begin{pmatrix}1&0\\0&0\end{pmatrix}},\,{\begin{pmatrix}0&1\\0&0\end{pmatrix}},\,{\begin{pmatrix}0&0\\1&0\end{pmatrix}},\,{\begin{pmatrix}0&0\\0&1\end{pmatrix}}\,.$

Show that

${}{\mathfrak {v}}={\begin{pmatrix}1&0\\0&1\end{pmatrix}},\,{\begin{pmatrix}1&0\\0&-1\end{pmatrix}},\,{\begin{pmatrix}0&1\\1&0\end{pmatrix}},\,{\begin{pmatrix}0&1\\-1&0\end{pmatrix}}\,$

is also a basis of ${}V$, and determine the transformation matrices.

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